open access

https://doi.org/10.14943/84131

https://hdl.handle.net/2115/69791

On possible isolated blow-up phenomena and regularity criterion of the 3D Navier-Stokes equation along the streamlines

英語

regularity criterion

2011-09-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

984

1-27

The first goal of our paper is to give a new type of regularity criterion for solutions u to Navier-Stokes equation in terms of some supercritical function space condition u 2 L1(L ; ) (with 3 4 (1712 􀀀 1) < < 3) and some exponential control on the growth rate of div( u juj ) along the streamlines of u. This regularity criterion greatly improves a previous result of the rst author. However, we also point out that totally new idea which involves the use of the new supercritical function space condition is necessary for the success of our new regularity criterion in this paper. The second goal of our paper is to construct a divergence free vector eld u within a owinvariant tubular region with increasing twisting of streamlines towards one end of a bundle of streamlines. The increasing twisting of streamlines is controlled in such a way that the associated quantities kukLp (2 < p < 3) and k div( u juj )kL6 blow up while preserving the nite energy property u 2 L2 at the same time. We also brie y mention how this construction is related to the regularity criterion proved in our paper.

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